You think you’re finally done writing a book! But in the case of A Guide to Plane Algebraic Curves there were many nice ideas deserving a warmer and longer handshake. This post is devoted to exploring some of them. Here are some ideas from the book’s Chapter 1, “A Gallery of Algebraic Curves.”
Morphing Between Curves:
A curve in the real plane that’s algebraic is defined by p(x, y) = 0, where p is a real polynomial. For example, if p = x2 + y2 - 1, then p (x, y) = 0 defines a circle. Likewise, q = x2 - y2 describes the two crossing lines x = ±y. It’s easy to use these two polynomials to create a continuous morphing between the circle and the two lines – think of t as a time parameter and write tp + (1 - t)q. When t = 0 we see the two lines, and when t has increased to 1, we get the circle. However, this simple linear combination has a mind of its own, and will carry out the transformation in a very natural way, albeit often (usually?) not the way we would predict! Here’s an animation of the two lines transforming to a circle:
Seeing this animation lets us pick the brain of the linear combination. What is its philosophy in bridging the chasm between the lines and the circle? The two lines immediately morph into hyperbolas, all of them passing though the four points in which the two lines and circle intersect. (In general, every morph between two curves contains all the intersection points of any two curves – be they the beginning and ending curves, or any two intermediate curves.) As t increases, the hyperbolas straighten, becoming two vertical lines which then close up into ellipses that squash down, finally becoming a circle.
By the way, what do you think the curves do as t runs through the interval (-∞, 0)? Through (1, +∞)?
Here is the Maple code creating this animation:
animate(implicitplot, [t*(x^2+y^2-1) + (1-t)*(x^2 – y^2), x=-4..4, y=-4..4, grid=[81,81], scaling=constrained, axes=none], t=0..1, frames=25, trace=5);
It’s not hard to reverse-engineer most of the code. The parameter t can take on values in any real interval. Increasing 25 creates more shots, making for a smoother transition between the beginning and ending curves. Changing 5 in trace alters the number of morphs the routine leaves behind as t runs through its interval. You can replace (x^2+y^2-1) and (x^2 – y^2) by other polynomials to create an endless variety of examples.
CHALLENGE: Googling “Wikipedia list of curves” leads you to a generous collection of algebraic curves. For those whose equations are given in cartesian form, using code like above can lead to some real surprises. What’s the most interesting morphed curve you can find?
FURTHERMORE . . . you can design a huge variety of curves on your own. Take a look:
Software like Maple or Mathematica can plot an endless variety of curves, but of course the code expects an equation of the curve to be drawn. What none of these drawing programs can do is reverse the process. That is, from a sketch of a curve, produce a polynomial p(x, y) whose set of zeros closely approximates the sketch. The book gives a number of “designer principles” to help address the problem. We now state three of them. (For more of them, and further discussions, see the last section of Chapter 1.)
If p(x, y) defines one curve and q(x, y) defines another, then pq defines their union. For example x - y defines the line through the origin of slope 1, and x + y defines the one of slope -1. Their product is x2 - y2 and it defines the union of the lines – precisely the lines we met above! This principle is an extraordinarily powerful tool; you could use any finite product to create a finite union of hundreds of curves. As a simple example, (x2 + y 2 - 1) (x2 + y 2 - 4) defines two concentric circles of radii 1 and 2.
The “initial part” of a polynomial p – the polynomial consisting of all lowest-degree terms of p – defines the union of the lines tangent to p’s curve at the origin. For example, the polynomial x3 + x2 - y2 defines what we may call an ‘’alpha curve,” looking like this:
The initial part is x2 - y2, and this defines the two lines through the origin of slope ±1. They are the two lines tangent to the alpha curve, shown dotted in the picture.
If an algebraic curve defined by p(x, y) isn’t bounded in the real plane, we can make it so by adding xn + yn to p(x, y), where n is even and of degree greater than that of p. When we do this, all parts of the curve going off to infinity turn around and connect up. For example, y3 + x2 defines a real “cusp curve,” and it’s unbounded, with two “branches” each heading off toward infinity. But add x4 + y4 to get y3 + x2 + x4 + y4, and far away from the origin the fourth-degree part dominates, preventing the entire polynomial from attaining the value 0. The part of the curve near the origin isn’t affected very much because there, the fourth-degree part is small. Here’s a picture of the original cusp curve, alongside the “teardrop curve” that results after bounding it:
CHALLENGE: Select a number of lines through the origin that divide the plane into congruent pie-shapes. Multiply their defining equations together, and then bound the lines by adding an even-powered term of sufficiently high degree. You can create a variety of roses this way, except some of the curves turn out to look more like propellers than flowers having symmetrically-shaped petals! What can you do to insure that each petal is symmetric about an axis through the origin? What effect does adding even terms with much higher degree than the number of lines?